Outputting inequality with $e^x$

Question

I many books I can find inequality which estimates $e$: $$\left(1+\frac{1}{n}\right)^n \lt e \lt \left(1+\frac{1}{n}\right)^{n+1}$$

I am wondering if correct is also to write: $$\left(1+\frac{1}{n}\right)^{nx} \lt e^x \lt \left(1+\frac{1}{n}\right)^{x(n+1)}$$ or maybe I can just raise the parties to the power?

Or maybe is any similiar inequality for $e^x$ like this?

Answer 1

yes for $x>0$, because taking something to the power of $x$ is a strictly increasing function.

http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf

To address your comment: here are some useful inequalities about the e, it is on the left of the first page.